Will it converge?

Calculus Level 5

Let a series be defined, for some real value of $$a_0$$ and $$\alpha(>0)$$ as:

$\displaystyle \large a_n=a_{n-1}+\alpha \sin(a_{n-1})$

$X=\lim _{ n\rightarrow \infty }{ { a }_{ n } }$

Let $$S$$ be the sum of all the distinct values of $$X$$ as $$a_0$$ varies from $$[1,100]$$ and $$\alpha$$ varies over all positive reals.

Find $\left\lfloor 10^4S \right\rfloor$

Details and Assumptions:

1. $$X$$ must be finite.

2. $$\alpha$$ is a variable independent of $$a_0$$ and can take any positive value.

×